At the beginning of each run, the game informs you which variant of Seraph you have. Each one takes different planning. Each variant has 999 HP on Covenant 0, and deals 9x3 damage, or 15 if he's Patient.
Seraph the Chaste will halve your buff and debuff stacks to make it more about sheer damage output. In this fight, you will have to deal with regular spawns of Lightwings (3 HP/12 ATK) as well as Pyrewings. (120 HP/2 ATK with Sweep and Emberdrain) Any unit afflicted with Emberdrain will lower the ember gained that round. You'll want to keep your floors either fairly light, as in one tank and one damage, or self-sufficient, as in a Wrathful Prince getting enough kills that he heals damage with armor. Additionally, I strongly recommend taking Ember from one of the previous bosses to help mitigate Emberdrain.
Seraph the Temperant will Sap your units, which acts as a reverse Rage. Additionally, this fight has Shadewings (5 HP/15x2 ATK) as a regular opponent. You need to either be able to outbuff the sap penalties or rely more on spells for damage.
Seraph the Diligent will make the first spell of each round Consumed, so you will not see it again during the fight. This fight also includes multiple waves of two Gilded Wings. (160 HP/8 ATK) In this fight, you need a high damage output floor or floors early on to prevent the Pyre from getting overwhelmed early. If using spells, having a few throwaway spells or spells that are already Consumed is incredibly handy.
Seraph the Patient will attack throughout the fight and apply Melee Weakness to whoever he hits. Additionally, he gains 1 ATK from every spell cast on the same floor and 3 ATK from every unit deployed on the same floor. You will need strong tanks to withstand the consistent Melee Weakness stacks.
In most Covenant 0 scenarios, getting to the relentless phase with your floors intact is the trick, hence the planning above. Having strong backline clear through spells or sweep units while building up your tanks and/or damage will leave you in a strong position to tear through Seraph. Even with this, a mistake in the relentless phase can cost you hundreds in damage.
Once you get the final blow, the achievement unlocks immediately.